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Mirrors > Home > MPE Home > Th. List > engch | Structured version Visualization version GIF version |
Description: The property of being a GCH-set is a cardinal invariant. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
engch | ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ GCH ↔ 𝐵 ∈ GCH)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enfi 8176 | . . 3 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) | |
2 | sdomen1 8104 | . . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≺ 𝑥 ↔ 𝐵 ≺ 𝑥)) | |
3 | pwen 8133 | . . . . . . 7 ⊢ (𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ 𝒫 𝐵) | |
4 | sdomen2 8105 | . . . . . . 7 ⊢ (𝒫 𝐴 ≈ 𝒫 𝐵 → (𝑥 ≺ 𝒫 𝐴 ↔ 𝑥 ≺ 𝒫 𝐵)) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ≺ 𝒫 𝐴 ↔ 𝑥 ≺ 𝒫 𝐵)) |
6 | 2, 5 | anbi12d 747 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 → ((𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ↔ (𝐵 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐵))) |
7 | 6 | notbid 308 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → (¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ↔ ¬ (𝐵 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐵))) |
8 | 7 | albidv 1849 | . . 3 ⊢ (𝐴 ≈ 𝐵 → (∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ↔ ∀𝑥 ¬ (𝐵 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐵))) |
9 | 1, 8 | orbi12d 746 | . 2 ⊢ (𝐴 ≈ 𝐵 → ((𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)) ↔ (𝐵 ∈ Fin ∨ ∀𝑥 ¬ (𝐵 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐵)))) |
10 | relen 7960 | . . . 4 ⊢ Rel ≈ | |
11 | 10 | brrelexi 5158 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V) |
12 | elgch 9444 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))) | |
13 | 11, 12 | syl 17 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))) |
14 | 10 | brrelex2i 5159 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
15 | elgch 9444 | . . 3 ⊢ (𝐵 ∈ V → (𝐵 ∈ GCH ↔ (𝐵 ∈ Fin ∨ ∀𝑥 ¬ (𝐵 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐵)))) | |
16 | 14, 15 | syl 17 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐵 ∈ GCH ↔ (𝐵 ∈ Fin ∨ ∀𝑥 ¬ (𝐵 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐵)))) |
17 | 9, 13, 16 | 3bitr4d 300 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ GCH ↔ 𝐵 ∈ GCH)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∀wal 1481 ∈ wcel 1990 Vcvv 3200 𝒫 cpw 4158 class class class wbr 4653 ≈ cen 7952 ≺ csdm 7954 Fincfn 7955 GCHcgch 9442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-1o 7560 df-2o 7561 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-gch 9443 |
This theorem is referenced by: gch2 9497 |
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